The smaller the stone, the faster it will probably pass. If it's a uric acid stone (gout), you can sometimes help to melt them with potassium citrate or lemonade. If it's calcium, they say that doesn't help as much. You didn't really say what kind of stone it was. All of them are supposed to be helped with plenty of fluids to increase solubility and prevent more growth. Usually you want a low sodium diet because that passes calcium and contributes more super saturation.
Infection struvite stones are sometimes kind of irregular shaped and are supposed to be problematic because they often staghorn into the kidney where they don't pass, so I'm assuming your doctors advice was to let this pass. so it's probalby calcium or uric acid.
I had one before and it passed when I drank lots of fluids. There is some theory that more fluids push on the stone and could cause more symptoms. One study suggested they pass either way, but I lean more towards having more fluids because it seemed to help me.
You definitely want to be hydrated though, because dehydration causes more stone growth.
The symptoms are caused by the stone so you're going to have to work on the pain managment and hydration. The alternative is pushing large instrumentation into your ureter which they usually do not do unless it's greater than 7mm or you have a risky kidney situation, like solitary kidney, diabetes, etc.
It's so invasive they have to leave a stent in place for weeks so your ureter doesn't not colapse, and stents usually cause pain, infection, bleeding and incontenance that usually resolves after removal.
As for The urethra. Entering in a male is tighter so it can be profoundly painful and damaging to the urethra, especially in younger patients.